#423655

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,16)(8,10)(9,22)(11,15)(12,18)(13,25)(14,20)(17,29)(19,31)(21,33)(23,27)(24,30)(26,28)(32,35)(34,36);(5,7)(6,12)(8,16)(9,23)(10,26)(11,19)(13,29)(14,31)(15,28)(17,24)(18,20)(21,34)(22,35)(25,36)(27,30)(32,33);(1,2,3,4)(5,8,20,15,11,14,10,6)(7,16,26,12,19,31,18,28)(9,21,33,22,25,30,24,13)(17,29,36,27,32,35,23,34);(5,9,10,24,11,25,20,33)(6,13,14,30,15,22,8,21)(7,17,18,23,19,32,26,36)(12,27,16,29,28,34,31,35)

Polycyclic presentation:

2^9|c^2=fi,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330851);